A321605 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(5*n) - A(x))^n.

1, 5, 35, 1110, 61830, 4607001, 422112085, 45521033720, 5625206604320, 782244114339935, 120812011501389376, 20514224767917807795, 3798925417133114909240, 762102329400356260363990, 164678708686403817727101920, 38140958485665617437764886383, 9427520984195812306085385378080, 2477372683628569966077893189614835, 689743886246438120027048924784220410

Offset: 0

Author

Paul D. Hanna, Nov 14 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 5*x + 35*x^2 + 1110*x^3 + 61830*x^4 + 4607001*x^5 + 422112085*x^6 + 45521033720*x^7 + 5625206604320*x^8 + 782244114339935*x^9 + ...
such that
1 = 1  +  ((1+x)^5 - A(x))  +  ((1+x)^10 - A(x))^2  +  ((1+x)^15 - A(x))^3  +  ((1+x)^20 - A(x))^4  +  ((1+x)^25 - A(x))^5  +  ((1+x)^30 - A(x))^6  +  ((1+x)^35 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)^5/(1 + (1+x)^5*A(x))^2  +  (1+x)^20/(1 + (1+x)^10*A(x))^3  +  (1+x)^45/(1 + (1+x)^15*A(x))^4  +  (1+x)^80/(1 + (1+x)^20*A(x))^5  +  (1+x)^125/(1 + (1+x)^25*A(x))^6  + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 5 + 45*x + 2930*x^2 + 225545*x^3 + 21445630*x^4 + 2388480630*x^5 + 303204843520*x^6 + 43104182972905*x^7 + 6777636393880895*x^8 + ...
the coefficients of which are all divisible by 5:
(1/5) * A'(x)/A(x) = 1 + 9*x + 586*x^2 + 45109*x^3 + 4289126*x^4 + 477696126*x^5 + 60640968704*x^6 + 8620836594581*x^7 + 1355527278776179*x^8 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A303056, A321602, A321603, A321604.

Cf. A326265.

Programs

PARI

{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(5*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} ((1+x)^(5*n) - A(x))^n.

(2) 1 = Sum_{n>=0} (1+x)^(5*n^2) / (1 + (1+x)^(5*n)*A(x))^(n+1).